State Variable Models

”State Variable Models

State variable models are made of 1st order ODEs instead of higher order equations.

They allow linear algebra to be used

S.V models are formed by defining higher order derivates as a cascade of 1st order derivatives.

Let’s define some variables…
$x_1\left(t\right)=x\left(t\right)$
$x_2\left(t\right)=x_1^{\prime }\left(t\right)=x^{\prime }\left(t\right)$ $x_3\left(t\right)=x_2^{\prime }\left(t\right)=x^{{}^{\prime \prime }}\left(t\right)$ And this trend continues, as we can see, higher order derivatives can be represented as a first order derivative of another variable…

A state variable model is basically an equation of motion using these state variable instead of regular variables

Your state variable model will look something like this:
$x_1^{\prime }=x_2\left(t\right)$ $x_2^{\prime }=x_3\left(t\right)$
$x_3^{\prime }\left(t\right)=\frac{f\left(t\right)}{a_3}-\frac{a_2}{a_3}{x}_3\left(t\right)-\frac{a_1}{a_3}{x}_2\left(t\right)-\frac{a_0}{a_3}{x}_1\left(t\right)$
Where you put this into a matrix modelling all three equations as described below.

Procedure

After creating a state variable model, you put it in matrix form where your state variable first derivatives are in a vector on one side, and your coefficients multiplied by the state variable vector is on the other side.

The Standard State Variable Model
$\left\{x^{\prime }\right\}=\left[A\right]\left\{x\right\}+\left[B\right]\left\{u\right\}$
Where:

• A: State matrix
• x: State vector
• B: input (control) matrix
• u: Input vector

The output form is basically the same except with output state matrices and output vectors…

You can also put an equation of motion into the LaPlace domain and then represent that as a state variable model and then represent it in matrix form.

Solution Method

• Isolate the “1” term in denominator
• Find the output of the integrator and call it a state (where in LaPlace the integrator is 1/s and the output term it is attached to)

Related

• State Space Model of Circuits